Coupled Pendula
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Abstract
A one dimensional oscillation can be thought of as an object moving back and forth about an equilibrium position. In this experiment we consider the oscillating system of two coupled pendulums (they are coupled by springs). The spring constants of each of the springs are determined and the odd and even mode frequencies were measured and calculated. Also, the beat frequency values were measured and calculated.
Introduction
The main purpose of this experiment was to analyse different oscillating systems, which are represented by pendulums that were set-up in a manner to produce even, odd and beat frequencies. The oscillating systems consist of two thin rods connected to a block of mass. The manner in which these rods are coupled is by springs. The springs, for example, can mimic the bonds between atomic nuclei. Different springs with different spring constants will be used. Further, the spring constants for each of the four springs will be calculated since the springs are a key feature of the experiment. Once the spring constant and frequencies for the various instances have been measured it is imperative to understand the behaviour of the springs. This is done by plotting various relationships, such as, ω² vs. l² and ω² vs. k where l is defined as the length from the top of the rod to a specific point on the rod where the spring is attached.
Theory
The main idea is to derive an expression for the angular frequency of the oscillating system in terms of the mass (m), the spring constant (k), length from the top of the rod to a position on the rod (l) and the total length (L). The general equation of motion of a simple pendulum is given as:
1
Now if we consider the fact that the string has no mass and the mass on the pendulum is perfectly spherical we can rewrite the equation as follows:
2
For the term we can use the small angle approximation:
3
By solving (3) we arrive at
4
Now consider two pendulums that are coupled; the restoring torque represented by:
5
If we choose to neglect the mass of the spring then pendulum 1 can be represented as:
6
Pendulum 2 has a similar representation. Now if we use the small angle approximation again we can solve for and
7a
7b
Once the initial conditions for odd modes are applied we obtain
8
We have now arrived at a working base. Equation (8) is what will be used to determine the theoretical values of odd mode frequencies. Once the binomial approximation is applied to equation (8) we arrive at an expression for the angular frequency
9
Moreover, the beat frequency is given by the difference of the angular frequency and :
10
Procedure
The procedure for this experiment can be found in the Physics 360A lab handout, titled “E5 Coupled Oscillators”.
Results
Spring Constants:
With the obtained data the first calculation that was made was that for the spring constant for each of the springs. The manner in which the data was obtained for this part was such that the springs were suspended vertically and then masses were added onto the springs (masses were added in increments of 10 grams). The displacement from the initial position to the final position was observed and recorded. The spring constant, k, was determined by graphing F(x) vs. x; where F(x) is the force exerted on the spring in Newtons (N) and x is the displacement in meters (m). The slope of this graph was then tabulated (the graph was linear so the slope calculation was quite simple: m = (y2 – y1)/(x2 – x1)).
Spring # | k (N/m) |
Uncertainty (± N/m %) |
1 |
29.96 |
2.190 |
2 |
21.07 |
1.600 |
3 |
4.745 |
0.6050 |
4 |
3.525 |
0.5610 |
Odd Modes
There were sixteen different values that were accounted for in the measurement of the odd mode frequencies. Different combinations of springs were used as well as different lengths. As already mentioned there were two plots that were made such as ω² vs. l² and ω² vs. k. These graphs help us understand the behaviour of the springs. The ω² vs. l² graph depicts the system where the spring constant is held constant and the length at which the springs are coupled were varied. Further, the graph of ω² vs. k depicts the system where the length is held constant and various springs with different springs constants are placed. The quantity appears in the equation for the angular frequency. Now is the natural frequency and this can be measured from the graph of ω² vs. k by extrapolating the y-intercept. Once this quantity has been measured it can be compared to the theoretical value. A table of the tabulated results are as follows:
Intercept values for ω² vs. k |
|||||
Length |
ωo² (rad²/s²) |
Uncertainty (± rad²/s²) |
Theoretical ωo² |
Uncertainty (± rad²/s²) |
% Diff. |
1 |
17.42 |
0.6671 |
17.37 |
0.308 |
0.29% |
2 |
17.43 |
0.6741 |
17.37 |
0.308 |
0.30% |
3 |
17.48 |
0.6674 |
17.37 |
0.308 |
0.31% |
4 |
17.47 |
0.6645 |
17.37 |
0.308 |
0.31% |
The information contained within the graph is quite interesting. The slope of this graph informs us of the effects that the different spring constants will have on the angular frequency. The theoretical value of the slope is 2l²/mL². The tabulated results for the slope are:
Slope Values for ω² vs. k |
|||||
Length |
Slope |
Uncertainty |
Theoretical 2l²/mL² |
Uncertainty |
% Diff. |
1 |
0.03350 |
0.0008549 |
0.03460 |
0.0001975 |
3.18% |
2 |
0.09010 |
0.002180 |
0.09824 |
0.0005305 |
9.03% |
3 |
0.1763 |
0.004073 |
0.1931 |
0.001006 |
8.70% |
4 |
0.3408 |
0.006475 |
0.3677 |
0.001787 |
7.32% |
Let us consider now the graph of ω² vs. l². Since the spring is held constant in this case and the length is varied we can see the effects of this in the change of the angular frequency. Here are the results:
Intercept values for ω² vs. l² |
|||||
Spring |
ωo² (rad²/s²) |
Uncertainty (± rad²/s²) |
Theoretical ωo² |
Uncertainty (± rad²/s²) |
% Diff. |
1 |
17.41 |
0.6731 |
17.37 |
0.308 |
0.23% |
2 |
17.42 |
0.6733 |
17.37 |
0.308 |
0.26% |
3 |
17.50 |
0.6770 |
17.37 |
0.308 |
0.74% |
4 |
17.55 |
0.6785 |
17.37 |
0.308 |
1.03% |
In this case the theoretical slope of the graph is given as 2k/mL². The slope gives us information that allows us to compare the change in the angular frequency upon changing the position of the spring. The results are as follows:
Slope Values for ω² vs. l² |
|||||
Spring |
2k/mL² |
Uncertainty |
Theoretical 2k/mL² |
Uncertainty |
% Diff. |
1 |
61.81 |
0.3307 |
63.78 |
1.406 |
3.09% |
2 |
42.28 |
0.2042 |
45.26 |
0.7268 |
6.58% |
3 |
10.23 |
0.03315 |
10.19 |
0.06306 |
0.391% |
4 |
7.656 |
0.02281 |
7.57 |
0.04351 |
1.12% |
Even Mode
The even modes of the system were measured with different springs at one length.
Beat Frequencies
In order to obtain a beat frequency one of the pendulums must be held motionless while the other one is set into motion. The neat thing is that the energy from the pendulum that was set into motion will be transferred to the pendulum that was motionless. In an ideal system this transfer of energy will continue on unless there is a force that stops it. But in our case the system is not ideal but the transfer of energy can be noticed. The time that it takes for the energy to transfer to the other pendulum is known as the beat frequency. The manner in which this is measured is measure the angular displacement and determining where the amplitude of the oscillations goes to zero. This occurs once every beat for each pendulum. The tabulated results are:
Spring | Measured Δω (rad/s) |
Theoretical Δω (rad/s) |
Uncertainty |
% Diff. |
1 |
0.3252 |
0.3496 |
0.000989 |
7.50% |
2 |
0.2490 |
0.2481 |
0.00174 |
0.361% |
Discussion
In the beginning of the experiment the spring constants were tabulated and showed that they had low uncertainty except for the 2.19% uncertainty for the first spring. Based on the table results for spring constants a relationship can be seen between the uncertainty and the strength of the spring. The spring which had a higher spring constant revealed a higher uncertainty while the springs which have a lower spring constant demonstrated a lower uncertainty since there was not enough force applied to extend them. Further, by extrapolating the graphs of ω² vs. l² and ω² vs. k the fundamental frequency was determined to be 17.46 rad/s ± 0.6712 rad/s. In this report the odd modes were studied first since the even mode frequencies were trivial. The frequencies of the odd mode oscillations were just the fundamental frequency. According to the theory, the odd mode frequencies were dependant on the springs used and the length at which they were placed. Moreover, the uncertainty in the beat frequencies was very low. The errors in most of the calculations are due to the fact that the set-up of the system is not ideal. Air resistance is present and this can account for the error in the data. However, the calculated values in all instances proved to be very close to the theoretical/predicted values. This proves that the theory of coupled oscillations is sound.
Conclusion
In conclusion, the spring constants were calculated to be 29.96 N/m ± 0.6561N/m; 21.07 N/m ± 0.3371 N/m; 4.745 ± 0.02871 N/m; 3.525 N/m ± 0.01978 N/m. The natural frequency of the pendulum was determined to be 17.46 rad/s ± 0.6712 rad/s. Moreover, the slopes of the graphs were on par with the theoretical values. Also, the measured value for the nautral frequency was very close to the theoretical value. Based on the theory, the odd mode frequencies as well as the beat frequencies were dependant the spring. This implies that the odd mode frequencies and beat frequencies will vary based on the spring that is used with the exception of the even modes. Moreover, this experiment has proved the fact that the theory of coupled oscillators can be used to model various other oscillating systems such as atoms and nuclei and radiation and absorption of energy.
References
1) “Experiment #5 Coupled Oscillators”, University of Waterloo, Physics 360A, 2008